Terminology
More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S×S to S:
Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure). If f is not a function, but is instead a partial function, it is called a partial operation. For instance, division of real numbers is a partial function, because one can't divide by zero: a/0 is not defined for any real a. Note however that both in algebra and model theory the binary operations considered are defined on all of S × S.
Sometimes, especially in computer science, the term is used for any binary function.
Binary operations are the keystone of algebraic structures studied in abstract algebra: they are essential in the definitions of groups, monoids, semigroups, rings, and more. Most generally, a magma is a set together with some binary operation defined on it.
Read more about this topic: Binary Operation